• Journal of the Korean Mathematical Society
• /
• v.55 no.5
• /
• pp.1091-1101
• /
• 2018

We concern in this paper the graph Kazdan-Warner equation $${\Delta}f=g-he^f$$ on an infinite graph, the prototype of which comes from the smooth Kazdan-Warner equation on an open manifold. Different from the variational methods often used in the finite graph case, we use a heat flow method to study the graph Kazdan-Warner equation. We prove the existence of a solution to the graph Kazdan-Warner equation under the assumption that $h{\leq}0$ and some other integrability conditions or constrictions about the underlying infinite graphs.

• Honam Mathematical Journal
• /
• v.43 no.2
• /
• pp.183-197
• /
• 2021

Recently several authors have extended the Beta function by using its integral representation. However, in many cases no expression of these extended functions in terms of classic special functions is known. In the present paper, we introduce a further extension by defining a family of functions G_r,s : ℝ^*₊ → ℂ, with r, s ∈ ℂ and ℜ(r) > 0. For given r, s, we prove that this function satisfies a second-order linear differential equation with rational coefficients. Solving this ODE, we express G_r,s as a combination of confluent hypergeometric functions. From this we deduce a new integral relation satisfied by Tricomi's function. We then investigate additional specific properties of G_r,1 which take the form of new non trivial integral relations involving exponential and error functions. We discuss the connection between G_r,1 and Stokes' first problem (or Rayleigh problem) in fluid mechanics which consists in determining the flow created by the movement of an infinitely long plate. For $r{\in}{\frac{1}{2}}{\mathbb{N}}^*$, we find additional relations between G_r,1 and Hermite polynomials. In view of these results, we believe the family of extended beta functions G_r,s will find further applications in two directions: (i) for improving our knowledge of confluent hypergeometric functions and Tricomi's function, (ii) and for engineering and physics problems.

• Geomechanics and Engineering
• /
• v.31 no.3
• /
• pp.291-304
• /
• 2022

Maximum shear modulus (G_max or G₀) is an important soil property useful for many engineering applications, such as the analysis of soil-structure interactions, soil stability, liquefaction evaluation, ground deformation and performance of seismic design. In the current study, bender element (BE) tests are used to evaluate the effect of the void ratio, effective confining pressure, grading characteristics (D₅₀, C_u and C_c), anisotropic consolidation and initial fabric anisotropy produced during specimen preparation on the G_max of sand-gravel mixtures. Based on the tests results, an empirical equation is proposed to predict G_max in granular soils, evaluated by the experimental data. The artificial neural network (ANN) and Adaptive Neuro Fuzzy Inference System (ANFIS) models were also applied. Coefficient of determination (R²) and Root Mean Square Error (RMSE) between predicted and measured values of G_max were calculated for the empirical equation, ANN and ANFIS. The results indicate that all methods accuracy is high; however, ANFIS achieves the highest accuracy amongst the presented methods.

• Kyungpook Mathematical Journal
• /
• v.47 no.1
• /
• pp.91-103
• /
• 2007

In this paper we prove the Hyers-Ulam-Rassias stability by considering the cases that the approximate remainder ${\varphi}$ is defined by $f(x{\ast}y)+f(x{\ast}y^{-1})-2g(x)-2g(y)={\varphi}(x,y)$, $f(x{\ast}y)+g(x{\ast}y^{-1})-2h(x)-2k(y)={\varphi}(x,y)$, where (G, *) is a group, X is a real or complex Hausdorff topological vector space and f, g, h, k are functions from G into X.

• Bulletin of the Korean Mathematical Society
• /
• v.61 no.4
• /
• pp.917-932
• /
• 2024

In this paper, we deal with the Euler-Maruyama (EM) scheme for stochastic differential equations driven by G-Brownian motion (G-SDEs). Under the linear growth and the local Lipschitz conditions, the strong convergence as well as the rate of convergence of the EM numerical solution to the exact solution for G-SDEs are established.

• East Asian mathematical journal
• /
• v.34 no.1
• /
• pp.69-84
• /
• 2018

In this paper, we study the viscoelastic Kirchhoff type equation with a nonlinear source for each independent kernels h and g with respect to Volterra terms. Under the smallness condition with respect to Kirchhoff coefficient and the relaxation function and other assumptions, we prove the uniform decay rate of the Kirchhoff type energy.

• Journal of applied mathematics & informatics
• /
• v.8 no.1
• /
• pp.45-57
• /
• 2001

A parametric scheme is proposed for the numerical solution of the nonlinear Boussinesq equation. The numerical method is developed by approximating the time and the space partical derivatives by finite-difference re placements and the nonlinear term by an appropriate linearized scheme. The resulting finite-difference method is analyzed for local truncation error and stability. The results of a number of numerical experiments are given for both the single and the double-soliton wave. AMS Mathematics Subject Classification : 65J15, 47H17, 49D15.

• Korean Journal of Mathematics
• /
• v.16 no.2
• /
• pp.205-214
• /
• 2008

In this paper, we prove the generalized Hyers-Ulam stability of a Cauchy-Jensen functional equation $2f(x+y,\frac{z+w}{2})=f(x,z)+f(x,w)+f(y,z)+f(y,w)$ in the spirit of $P.G{\breve{a}}vruta$.

• Journal of applied mathematics & informatics
• /
• v.13 no.1_2
• /
• pp.11-27
• /
• 2003

A fourth order in time and second order in space scheme using a finite-difference method is developed for the non-linear Boussinesq equation. For the solution of the resulting non-linear system a predictor-corrector pair is proposed. The method is analyzed for local truncation error and stability. The results of a number of numerical experiments for both the single and the double-soliton waves are given.

• Journal of applied mathematics & informatics
• /
• v.8 no.3
• /
• pp.683-691
• /
• 2001

A linearized finite-difference scheme is used to transform the initial/boundary-value problem associated with the nonlinear Schrodinger equation into a linear algebraic system. This method is developed by replacing the time and the nonlinear term by an appropriate parametric linearized scheme based on Taylor’s expansion. The resulting finite-difference method is analysed for stability and convergence. The results of a number of numerical experiments for the single-soliton wave are given.